On the zeros of the Riemann zeta function in the critical strip. III
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- by J. van de Lune and H. J. J. te Riele PDF
- Math. Comp. 41 (1983), 759-767 Request permission
Corrigendum: Math. Comp. 46 (1986), 771.
Abstract:
We describe extensive computations which show that Riemann’s zeta function $\zeta (s)$ has exactly 300,000,001 zeros of the form $\sigma + it$ in the region $0 < t < 119,590,809.282$. All these zeros are simple and lie on the line $\sigma = \frac {1}{2}$. (This extends a similar result for the first 200,000,001 zeros, established by Brent, van de Lune, te Riele and Winter in Math. Comp., v. 39, 1982, pp. 681-688.) Counts of the numbers of Gram blocks of various types and the failures of "Rosser’s rule" are given, together with some graphs of the function $Z(t)$ near the first observed failures of Rosser’s rule.References
- Richard P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), no. 148, 1361–1372. MR 537983, DOI 10.1090/S0025-5718-1979-0537983-2
- R. P. Brent, J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. II, Math. Comp. 39 (1982), no. 160, 681–688. MR 669660, DOI 10.1090/S0025-5718-1982-0669660-1
- E. Karkoschka and P. Werner, Einige Ausnahmen zur Rosserschen Regel in der Theorie der Riemannschen Zetafunktion, Computing 27 (1981), no. 1, 57–69 (German, with English summary). MR 623176, DOI 10.1007/BF02243438 J. van de Lune, H. J. J. te Riele & D. T. Winter, Rigorous High Speed Separation of Zeros of Riemann’s Zeta Function, Report NW 113/81, October 1981, Mathematical Centre, Amsterdam. J. van de Lune & H. J. J. te Riele, Rigorous High Speed Separation of Zeros of Riemann’s Zeta Function, II, Report NN 26/82, June 1982, Mathematical Centre, Amsterdam.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Math. Comp. 41 (1983), 759-767
- MSC: Primary 11M26; Secondary 11-04, 11Y35
- DOI: https://doi.org/10.1090/S0025-5718-1983-0717719-3
- MathSciNet review: 717719